Practical Geometry. construction of a Line ParaLLeL to a given Line through a Point not on it
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- Meryl McKenzie
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1 12 ractica Geometry introduction In the previous cass, you have earnt to construct a circe of given radius, a ine segment of given ength, a copy of a ine segment, a perpendicuar ine to a given ine at a point on it or outside it, perpendicuar bisector of a ine segment, an ange of given measure, a copy of an ange, bisector of an ange and anges of some specia measures 60, 30, 120, 90, 45 etc. In this chapter, we sha earn: construction of a ine parae to a given ine through a point not on the ine. construction of trianges. construction of a Line arallel to a given Line through a oint not on it We sha draw a ine parae to a given ine through a point not on it by the foowing methods: (i) by paper foding (activity) (ii) by using ruer and set square (iii) by using ruer and compasses. activity 11 To draw a ine parae to a given ine through a point not on it steps 1. Take a sheet (or piece) of tracing paper and draw a ine on it. Mark a point not on and a point on as shown in fig. (i). 2. Fod the paper through perpendicuar to and form a crease as shown in fig. (ii). 3. Unfod the paper and draw a dotted ine, say n, on the above fod as shown in fig. (iii). Materias required (i) Tracing paper (ii) ooured ba point pen (iii) uer. 4. Fod the paper perpendicuar to ine n and move this fod ti it passes through and form a crease as shown in fig. (iv). 5. Unfod the paper and draw a ine, say m, on this new crease as shown in fig. (v). This new ine m is the required ine parae to the given ine and passing through point.
2 236 Learning Mathematics VII (i) (ii) n n (iii) (iv) n m Using ruer and set square To draw a ine parae to a given ine through a point not on it. Given. ine and a point outside. equired. To draw a ine parae to and passing through the given point. (v) 1. ace your set square so that one of its shorter edge XY just ies aong the ine. x y z 2. ace your ruer so that one edge of the ruer just ies aong the other shorter edge XZ of the set square. Hod the ruer firmy and side the set square aong the ruer unti the edge XY of the set square passes through. x z y
3 ractica Geometry 237 x equired ine y 3. Draw the ine aong the edge XY of the set square. This is the required ine through which is parae to the ine. z Using ruer and compasses To draw a ine parae to a given ine through a point not on it. Given. ny ine and a point outside. equired. To draw a ine parae to and passing through the point. 1. Take any point Q on. Join and Q. F 2. With Q as centre and any suitabe radius, draw an arc to meet at and Q at D. E 3. With as centre and same radius (as in step 2), draw an arc to D meet Q at E. 4. Measure the segment D with compass. Q 5. With E as centre and radius equa to D, draw an arc to cut the previous arc at F. 6. Draw a ine passing through and F, then F is the required ine parae to the ine and passing through. emark Note that in the above construction, Q and QF are aternate interior anges. Therefore, ine and F are parae. Moreover, we can sighty modify the above construction to use the concept of equa corresponding anges instead of equa aternate interior anges. onstruction of Trianges eca, the sum of engths of any two sides of a triange is greater than the ength of the third side. so the sum of measures of a the three anges of a triange is 180. In the chapter on ongruence of Trianges, we noticed that a triange can be uniquey drawn if any of the foowing sets of measurements is given: (i) the engths of three sides of a triange. (ii) the engths of two sides and measure of incuded ange between them. (iii) two anges and the ength of the incuded side between them. (iv) the ength of hypotenuse and a side in case of a right anged triange. We sha use the above ideas to construct trianges.
4 238 Learning Mathematics VII onstructing a triange when the engths of its three sides are given Exampe 1. onstruct a triange Q such that Q = 5.6 cm, Q = 4.7 cm and = 3.4 cm. Soution. Draw a rough sketch of Q and mark the given engths (this wi hep us in deciding how to proceed) 3.4 cm 4.7 cm 5.6 cm Q 1. Draw a ine segment Q of ength 5.6 cm. 2 With Q as centre and radius 4.7 cm (= Q), draw an arc. 3. With as centre and radius 3.4 cm (= ), draw an arc to cut the previous arc at. 4. Join and Q. Then Q is the required triange. Exampe 2. onstruct an equiatera triange with side 4.5 cm. Soution. 1. Draw a ine segment of ength 4.5 cm. 2. With as centre and radius 4.5 cm, draw an arc. 3. With as centre and radius 4.5 cm, draw an arc to cut the previous arc at. 4. Joint and, then is the required equiatera triange with side 4.5 cm. 3.4 cm 5.6 cm 4.7 cm 4.5 cm 4.5 cm 4.5 cm Q onstructing a triange when the engths of its two sides and the measure of the incuded ange between them are known Exampe 3. onstruct a triange such that = 3.2 cm, = 3.9 cm and = 120. Soution. Draw a rough sketch of with measures marked on it. 3.2 cm cm 1. Draw a ine segment of ength 3.2 cm. 2. t, construct = With as centre and radius 3.9 cm, draw an arc to meet at cm 4. Join, then is the required triange cm Exampe 4. onstruct an isoscees triange in which ength of each equa side is and the ange between them is 80. Measure other anges. Soution. Draw a rough sketch of a triange with measures marked on it. 80
5 ractica Geometry Draw a ine segment of ength. 2. t, draw = 80 (by using protractor). 3. With as centre and radius draw an arc to meet at. 4. Join, then is the required isoscees triange with given measurements. On measuring and by protractor, we find that: = 50 and = onstruction of a triange when the measures of two of its anges and the ength of the side incuded between them are given Exampe 5. onstruct a triange such that = 6.3 cm, = 45 and = 60 by using ruer and compasses ony. Soution. Draw a rough sketch of with measures marked on it. 1. Draw a ine segment of ength 6.3 cm. 2. t, construct = t, construct Q = Let rays and Q intersect at, then is the required triange cm Q 6. 3 cm Exampe 6. onstruct Q if Q = 5 cm, m Q = 105 and m Q = 40. Soution. s the sum of anges of a triange is 180, \ Q + Q + Q = 180 Q = 180 Q = = 35. Draw a rough sketch of Q with Q = 5 cm, Q = 35 and Q = cm 40 Q T S 1. Draw a ine segment Q of ength 5 cm. 2. t, draw SQ = 35 (by using protractor). 3. t Q, draw QT = 40 (by using protractor). 4. Let rays S and QT meet at, then Q is the required triange with given data cm Q
6 240 Learning Mathematics VII onstruction of a right anged triange when the engths of hypotenuse and one side are given Exampe 7. onstruct a triange such that = 3.2 cm, = 90 and hypotenuse = 5.8 cm by using ruer and compasses ony. 5.8 cm Soution. Draw a rough sketch of with = 3.2 cm, = 90 and hypotenuse = 5.8 cm. 3.2 cm 1. Draw a ine segment of ength 3.2 cm. 2. t, construct = With as centre and radius 5.8 cm, draw an arc to meet at. 4. Join, then is the required triange. 5.8 cm cm Exercise Draw a ine, say, take a point outside it. Through, draw a ine parae to using ruer and compasses ony. 2. Draw a ine. Draw a perpendicuar to at any point on. On this perpendicuar choose a point, 3.5 cm away from ine. Through, draw a ine m parae to. 3. Let be a ine and be a point not on. Through, draw a ine m parae to. Now join to any point Q on. hoose any other point on m. Through, draw a ine parae to Q. If this ine meets at S, then what shape do the two sets of parae ines encose? 4. Draw any triange. Through, draw a ine parae to by using (i) ruer and compasses (ii) ruer and set square. 5. onstruct a triange, given that (i) = 5 cm, = 6 cm and = 7 cm (ii) = 4.5 cm, = 5 cm and = 6 cm. 6. onstruct a triange Q given that Q = 5.4 cm, Q = = 4.7 cm. Name the triange. 7. onstruct a triange LMN such that ength of each side is 5.3 cm. Name the triange. 8. onstruct a triange such that = 2.5 cm, = 6 cm and = 6.5 cm. Measure and name the triange. 9. onstruct a triange Q, given that Q = 3 cm, Q = and Q = onstruct DEF such that DE = 5 cm, DF = 3 cm and m EDF = onstruct a triange (using ruer and protractor) given that = 5.2 cm, = 58 and = 76. Measure. 12. onstruct an isoscees triange in which the ength of each of its equa sides is 6.5 cm and the ange between them is 110. Measure base anges. 13. onstruct triange XYZ if it is given that XY = 6 cm, X = 30 and Y = onstruct a triange Q given that Q = 4.9 cm, = 45 and Q = 60. Measure. 15. onstruct a triange Q, given that Q = 5 cm, m Q = 105 and m Q = 40. [Hint. m Q = = 35 ]
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